Logic for Computer Scientists/Modal Logic/Temporal Logics

The two modalities ◻{\displaystyle \square } and ⋄{\displaystyle \diamond } cannot be used to distinguish between past and future. For this we need a multi-modal logic with the following ◻{\displaystyle \square }-operators

The semantics is then given as before, by giving constraints for the three reachability relations or by giving appropriate axioms, e.g.

[F]A→[F][F]A{\displaystyle [F]A\to [F][F]A}: Transitivity; an analog axiom holds for the two other ◻{\displaystyle \square }-operators.

A→[F]⟨P⟩A{\displaystyle A\to [F]\langle P\rangle A}: if we go from a time point t{\displaystyle t} in the future t′{\displaystyle t'}, we can go back in the past to the time point where A{\displaystyle A} was true.